(* Title: HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML
Author: Jasmin Blanchette, TU Muenchen
Author: Steffen Juilf Smolka, TU Muenchen
*)
signature SLEDGEHAMMER_PROOF_RECONSTRUCT =
sig
type isar_params = ATP_Proof_Reconstruct.isar_params
type one_line_params = ATP_Proof_Reconstruct.one_line_params
val isar_proof_text :
Proof.context -> bool -> isar_params ->
one_line_params -> string
val proof_text :
Proof.context -> bool -> isar_params ->
int -> one_line_params -> string
end;
structure Sledgehammer_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =
struct
open ATP_Util
open ATP_Proof
open ATP_Problem_Generate
open ATP_Proof_Reconstruct
open String_Redirect
(** Type annotations **)
fun post_traverse_term_type' f _ (t as Const (_, T)) s = f t T s
| post_traverse_term_type' f _ (t as Free (_, T)) s = f t T s
| post_traverse_term_type' f _ (t as Var (_, T)) s = f t T s
| post_traverse_term_type' f env (t as Bound i) s = f t (nth env i) s
| post_traverse_term_type' f env (Abs (x, T1, b)) s =
let
val ((b', s'), T2) = post_traverse_term_type' f (T1 :: env) b s
in f (Abs (x, T1, b')) (T1 --> T2) s' end
| post_traverse_term_type' f env (u $ v) s =
let
val ((u', s'), Type (_, [_, T])) = post_traverse_term_type' f env u s
val ((v', s''), _) = post_traverse_term_type' f env v s'
in f (u' $ v') T s'' end
fun post_traverse_term_type f s t =
post_traverse_term_type' (fn t => fn T => fn s => (f t T s, T)) [] t s |> fst
fun post_fold_term_type f s t =
post_traverse_term_type (fn t => fn T => fn s => (t, f t T s)) s t |> snd
(* Data structures, orders *)
val cost_ord = prod_ord int_ord (prod_ord int_ord int_ord)
structure Var_Set_Tab = Table(
type key = indexname list
val ord = list_ord Term_Ord.fast_indexname_ord)
(* (1) Generalize Types *)
fun generalize_types ctxt t =
t |> map_types (fn _ => dummyT)
|> Syntax.check_term
(Proof_Context.set_mode Proof_Context.mode_pattern ctxt)
(* (2) Typing-spot Table *)
local
fun key_of_atype (TVar (idxn, _)) =
Ord_List.insert Term_Ord.fast_indexname_ord idxn
| key_of_atype _ = I
fun key_of_type T = fold_atyps key_of_atype T []
fun update_tab t T (tab, pos) =
(case key_of_type T of
[] => tab
| key =>
let val cost = (size_of_typ T, (size_of_term t, pos)) in
case Var_Set_Tab.lookup tab key of
NONE => Var_Set_Tab.update_new (key, cost) tab
| SOME old_cost =>
(case cost_ord (cost, old_cost) of
LESS => Var_Set_Tab.update (key, cost) tab
| _ => tab)
end,
pos + 1)
in
val typing_spot_table =
post_fold_term_type update_tab (Var_Set_Tab.empty, 0) #> fst
end
(* (3) Reverse-Greedy *)
fun reverse_greedy typing_spot_tab =
let
fun update_count z =
fold (fn tvar => fn tab =>
let val c = Vartab.lookup tab tvar |> the_default 0 in
Vartab.update (tvar, c + z) tab
end)
fun superfluous tcount =
forall (fn tvar => the (Vartab.lookup tcount tvar) > 1)
fun drop_superfluous (tvars, (_, (_, spot))) (spots, tcount) =
if superfluous tcount tvars then (spots, update_count ~1 tvars tcount)
else (spot :: spots, tcount)
val (typing_spots, tvar_count_tab) =
Var_Set_Tab.fold
(fn kv as (k, _) => apfst (cons kv) #> apsnd (update_count 1 k))
typing_spot_tab ([], Vartab.empty)
|>> sort_distinct (rev_order o cost_ord o pairself snd)
in fold drop_superfluous typing_spots ([], tvar_count_tab) |> fst end
(* (4) Introduce Annotations *)
fun introduce_annotations thy spots t t' =
let
val get_types = post_fold_term_type (K cons) []
fun match_types tp =
fold (Sign.typ_match thy) (op ~~ (pairself get_types tp)) Vartab.empty
fun unica' b x [] = if b then [x] else []
| unica' b x (y :: ys) =
if x = y then unica' false x ys
else unica' true y ys |> b ? cons x
fun unica ord xs =
case sort ord xs of x :: ys => unica' true x ys | [] => []
val add_all_tfree_namesT = fold_atyps (fn TFree (x, _) => cons x | _ => I)
fun erase_unica_tfrees env =
let
val unica =
Vartab.fold (add_all_tfree_namesT o snd o snd) env []
|> unica fast_string_ord
val erase_unica = map_atyps
(fn T as TFree (s, _) =>
if Ord_List.member fast_string_ord unica s then dummyT else T
| T => T)
in Vartab.map (K (apsnd erase_unica)) env end
val env = match_types (t', t) |> erase_unica_tfrees
fun get_annot env (TFree _) = (false, (env, dummyT))
| get_annot env (T as TVar (v, S)) =
let val T' = Envir.subst_type env T in
if T' = dummyT then (false, (env, dummyT))
else (true, (Vartab.update (v, (S, dummyT)) env, T'))
end
| get_annot env (Type (S, Ts)) =
(case fold_rev (fn T => fn (b, (env, Ts)) =>
let
val (b', (env', T)) = get_annot env T
in (b orelse b', (env', T :: Ts)) end)
Ts (false, (env, [])) of
(true, (env', Ts)) => (true, (env', Type (S, Ts)))
| (false, (env', _)) => (false, (env', dummyT)))
fun post1 _ T (env, cp, ps as p :: ps', annots) =
if p <> cp then
(env, cp + 1, ps, annots)
else
let val (_, (env', T')) = get_annot env T in
(env', cp + 1, ps', (p, T') :: annots)
end
| post1 _ _ accum = accum
val (_, _, _, annots) = post_fold_term_type post1 (env, 0, spots, []) t'
fun post2 t _ (cp, annots as (p, T) :: annots') =
if p <> cp then (t, (cp + 1, annots))
else (Type.constraint T t, (cp + 1, annots'))
| post2 t _ x = (t, x)
in post_traverse_term_type post2 (0, rev annots) t |> fst end
(* (5) Annotate *)
fun annotate_types ctxt t =
let
val thy = Proof_Context.theory_of ctxt
val t' = generalize_types ctxt t
val typing_spots =
t' |> typing_spot_table
|> reverse_greedy
|> sort int_ord
in introduce_annotations thy typing_spots t t' end
fun string_for_proof ctxt type_enc lam_trans i n =
let
fun fix_print_mode f x =
Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)
(print_mode_value ())) f x
fun do_indent ind = replicate_string (ind * indent_size) " "
fun do_free (s, T) =
maybe_quote s ^ " :: " ^
maybe_quote (fix_print_mode (Syntax.string_of_typ ctxt) T)
fun do_label l = if l = no_label then "" else string_for_label l ^ ": "
fun do_have qs =
(if member (op =) qs Moreover then "moreover " else "") ^
(if member (op =) qs Ultimately then "ultimately " else "") ^
(if member (op =) qs Then then
if member (op =) qs Show then "thus" else "hence"
else
if member (op =) qs Show then "show" else "have")
val do_term =
maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)
o annotate_types ctxt
val reconstr = Metis (type_enc, lam_trans)
fun do_facts (ls, ss) =
reconstructor_command reconstr 1 1 [] 0
(ls |> sort_distinct (prod_ord string_ord int_ord),
ss |> sort_distinct string_ord)
and do_step ind (Fix xs) =
do_indent ind ^ "fix " ^ space_implode " and " (map do_free xs) ^ "\n"
| do_step ind (Let (t1, t2)) =
do_indent ind ^ "let " ^ do_term t1 ^ " = " ^ do_term t2 ^ "\n"
| do_step ind (Assume (l, t)) =
do_indent ind ^ "assume " ^ do_label l ^ do_term t ^ "\n"
| do_step ind (Prove (qs, l, t, By_Metis facts)) =
do_indent ind ^ do_have qs ^ " " ^
do_label l ^ do_term t ^ " " ^ do_facts facts ^ "\n"
| do_step ind (Prove (qs, l, t, Case_Split (proofs, facts))) =
implode (map (prefix (do_indent ind ^ "moreover\n") o do_block ind)
proofs) ^
do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^ " " ^
do_facts facts ^ "\n"
and do_steps prefix suffix ind steps =
let val s = implode (map (do_step ind) steps) in
replicate_string (ind * indent_size - size prefix) " " ^ prefix ^
String.extract (s, ind * indent_size,
SOME (size s - ind * indent_size - 1)) ^
suffix ^ "\n"
end
and do_block ind proof = do_steps "{ " " }" (ind + 1) proof
(* One-step proofs are pointless; better use the Metis one-liner
directly. *)
and do_proof [Prove (_, _, _, By_Metis _)] = ""
| do_proof proof =
(if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^
do_indent 0 ^ "proof -\n" ^ do_steps "" "" 1 proof ^ do_indent 0 ^
(if n <> 1 then "next" else "qed")
in do_proof end
fun min_local ctxt type_enc lam_trans proof =
let
(* Merging spots, greedy algorithm *)
fun cost (Prove (_, _ , t, _)) = Term.size_of_term t
| cost _ = ~1
fun can_merge (Prove (_, lbl, _, By_Metis _)) (Prove (_, _, _, By_Metis _)) =
(lbl = no_label)
| can_merge _ _ = false
val merge_spots =
fold_index
(fn (i, s2) => fn (s1, pile) => (s2, pile |> can_merge s1 s2 ? cons (i, cost s1)))
(tl proof) (hd proof, [])
|> snd |> sort (rev_order o int_ord o pairself snd) |> map fst
(* Enrich context with facts *)
val thy = Proof_Context.theory_of ctxt
fun sorry t = Skip_Proof.make_thm thy (HOLogic.mk_Trueprop t) (* FIXME: mk_Trueprop always ok? *)
fun enrich_ctxt' (Prove (_, lbl, t, _)) ctxt =
if lbl = no_label then ctxt
else Proof_Context.put_thms false (string_for_label lbl, SOME [sorry t]) ctxt
| enrich_ctxt' _ ctxt = ctxt
val rich_ctxt = fold enrich_ctxt' proof ctxt
(* Timing *)
fun take_time tac arg =
let
val t_start = Timing.start ()
in
(tac arg ; Timing.result t_start |> #cpu)
end
fun try_metis (Prove (qs, _, t, By_Metis fact_names)) s0 =
let
fun thmify (Prove (_, _, t, _)) = sorry t
val facts = fact_names |>> map string_for_label
|> op@
|> map (Proof_Context.get_thm rich_ctxt)
|> (if member op= qs Then
then cons (the s0 |> thmify)
else I)
val goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop t) (* FIXME: mk_Trueprop always ok? *)
fun tac {context = ctxt, prems = _} =
Metis_Tactic.metis_tac [type_enc] lam_trans ctxt facts 1
in
take_time (fn () => goal tac)
end
(* Merging *)
fun merge (Prove (qs1, _, _, By_Metis (ls1, ss1)))
(Prove (qs2, lbl , t, By_Metis (ls2, ss2))) =
let
val qs = (inter op= qs1 qs2) (* FIXME: Is this correct? *)
|> member op= (union op= qs1 qs2) Ultimately ? cons Ultimately
|> member op= qs2 Show ? cons Show
in Prove (qs, lbl, t, By_Metis (ls1@ls2, ss1@ss2)) end
fun try_merge proof i =
let
val (front, s0, s1, s2, tail) =
case (proof, i) of
((s1::s2::proof), 0) => ([], NONE, s1, s2, proof)
| _ => let val (front, s0::s1::s2::tail) = chop (i-1) proof
in (front, SOME s0, s1, s2, tail) end
val s12 = merge s1 s2
val t1 = try_metis s1 s0 ()
val t2 = try_metis s2 (SOME s1) ()
val tlimit = t1 + t2 |> Time.toReal |> curry Real.* 1.2 |> Time.fromReal
in
(TimeLimit.timeLimit tlimit (try_metis s12 s0) ();
SOME (front @ (case s0 of NONE => s12::tail | SOME s => s::s12::tail)))
handle _ => NONE
end
fun merge_steps proof [] = proof
| merge_steps proof (i::is) =
case try_merge proof i of
NONE => merge_steps proof is
| SOME proof' => merge_steps proof' (map (fn j => if j>i then j-1 else j) is)
in merge_steps proof merge_spots end
fun isar_proof_text ctxt isar_proof_requested
(debug, isar_shrink_factor, pool, fact_names, sym_tab, atp_proof, goal)
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
val isar_shrink_factor =
(if isar_proof_requested then 1 else 2) * isar_shrink_factor
val (params, hyp_ts, concl_t) = strip_subgoal ctxt goal subgoal
val frees = fold Term.add_frees (concl_t :: hyp_ts) []
val one_line_proof = one_line_proof_text 0 one_line_params
val type_enc =
if is_typed_helper_used_in_atp_proof atp_proof then full_typesN
else partial_typesN
val lam_trans = lam_trans_from_atp_proof atp_proof metis_default_lam_trans
fun isar_proof_of () =
let
val atp_proof =
atp_proof
|> clean_up_atp_proof_dependencies
|> nasty_atp_proof pool
|> map_term_names_in_atp_proof repair_name
|> decode_lines ctxt sym_tab
|> rpair [] |-> fold_rev (add_line fact_names)
|> repair_waldmeister_endgame
|> rpair [] |-> fold_rev add_nontrivial_line
|> rpair (0, [])
|-> fold_rev (add_desired_line isar_shrink_factor fact_names frees)
|> snd
val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)
val conjs =
atp_proof
|> map_filter (fn Inference_Step (name as (_, ss), _, _, []) =>
if member (op =) ss conj_name then SOME name else NONE
| _ => NONE)
fun dep_of_step (Definition_Step _) = NONE
| dep_of_step (Inference_Step (name, _, _, from)) = SOME (from, name)
val ref_graph = atp_proof |> map_filter dep_of_step |> make_ref_graph
val axioms = axioms_of_ref_graph ref_graph conjs
val tainted = tainted_atoms_of_ref_graph ref_graph conjs
val props =
Symtab.empty
|> fold (fn Definition_Step _ => I (* FIXME *)
| Inference_Step ((s, _), t, _, _) =>
Symtab.update_new (s,
t |> fold forall_of (map Var (Term.add_vars t []))
|> member (op = o apsnd fst) tainted s ? s_not))
atp_proof
fun prop_of_clause c =
fold (curry s_disj) (map_filter (Symtab.lookup props o fst) c)
@{term False}
fun label_of_clause [name] = raw_label_for_name name
| label_of_clause c = (space_implode "___" (map fst c), 0)
fun maybe_show outer c =
(outer andalso length c = 1 andalso subset (op =) (c, conjs))
? cons Show
fun do_have outer qs (gamma, c) =
Prove (maybe_show outer c qs, label_of_clause c, prop_of_clause c,
By_Metis (fold (add_fact_from_dependency fact_names
o the_single) gamma ([], [])))
fun do_inf outer (Have z) = do_have outer [] z
| do_inf outer (Hence z) = do_have outer [Then] z
| do_inf outer (Cases cases) =
let val c = succedent_of_cases cases in
Prove (maybe_show outer c [Ultimately], label_of_clause c,
prop_of_clause c,
Case_Split (map (do_case false) cases, ([], [])))
end
and do_case outer (c, infs) =
Assume (label_of_clause c, prop_of_clause c) ::
map (do_inf outer) infs
val isar_proof =
(if null params then [] else [Fix params]) @
(ref_graph
|> redirect_graph axioms tainted
|> chain_direct_proof
|> map (do_inf true)
|> kill_duplicate_assumptions_in_proof
|> kill_useless_labels_in_proof
|> relabel_proof
|> min_local ctxt type_enc lam_trans)
|> string_for_proof ctxt type_enc lam_trans subgoal subgoal_count
in
case isar_proof of
"" =>
if isar_proof_requested then
"\nNo structured proof available (proof too short)."
else
""
| _ =>
"\n\n" ^ (if isar_proof_requested then "Structured proof"
else "Perhaps this will work") ^
":\n" ^ Markup.markup Isabelle_Markup.sendback isar_proof
end
val isar_proof =
if debug then
isar_proof_of ()
else case try isar_proof_of () of
SOME s => s
| NONE => if isar_proof_requested then
"\nWarning: The Isar proof construction failed."
else
""
in one_line_proof ^ isar_proof end
fun proof_text ctxt isar_proof isar_params num_chained
(one_line_params as (preplay, _, _, _, _, _)) =
(if case preplay of Failed_to_Play _ => true | _ => isar_proof then
isar_proof_text ctxt isar_proof isar_params
else
one_line_proof_text num_chained) one_line_params
end;